Scrambling and entanglement spreading in long-range spin chains
Silvia Pappalardi,1, 2,∗ Angelo Russomanno,3, 2 Bojan ˇZunkoviˇc,4Fernando Iemini,2, 5 Alessandro Silva,1 and Rosario Fazio2, 3
1SISSA, Via Bonomea 265, I-34135 Trieste, Italy 2Abdus Salam ICTP, Strada Costiera 11, I-34151 Trieste, Italy 3
NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
4Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia 5Instituto de Fisica, Universidade Federal Fluminense, 24210-346 Niteroi, Brazil
We study scrambling in connection to multipartite entanglement dynamics in regular and chaotic long-range spin chains, characterized by a well defined semi-classical limit. For regular dynamics, scrambling and entanglement dynamics are found to be very different: up to the Ehrenfest time, they rise side by side departing only afterward. Entanglement saturates and becomes extensively multipartite, while scrambling, characterized by the dynamic of the square commutator of initially commuting variables, continues its growth up to the recurrence time. Remarkably, the exponential growth of the latter emerges not only in the chaotic case but also in the regular one, when the dynamics occurs at a dynamical critical point.
I. INTRODUCTION
Classical systems with long-range interactioruns dis-play many interesting dynamical properties that have been extensively studied since many decades1. In the
quantum domain, instead, long-range systems have been the focus of a great deal of attention only lately, as a result of their experimental simulation with different platforms2–5. These systems allow the controlled study
of quantum dynamics in the absence of significant deco-herence, a property that allows the study of a number of important phenomena as, for example, dynamical phase transitions6–8 or the dynamics of correlations9–13 in a
situation where Lieb-Robinson bounds do not apply14,15.
It is well established that understanding the coherent dynamics of a quantum many-body system requires a thorough understanding of the behaviour of its quantum correlations16,17. The spreading of quantum correlations
has been the focus of a lot of theoretical efforts18,
start-ing from the initial important results on the dynamics of entanglement entropy19. Very recently a new way to
characterise quantum dynamics of many-body systems has been proposed, based on the concept of scrambling. Initially introduced as a probe of quantum chaos20–23,
scrambling is generically identified as the delocalisation of quantum information24 in a many-body system. A
measure of scrambling is associated with the growth of the square commutator between two initially commuting observables20,21. For quantum chaotic systems20,21,25,26,
this quantity is expected to grow exponentially before the Ehrenfest time - defined as the time at which semi-classics breaks and quantum effect become dominant56 -otherwise, it grows at most polynomially in time27–29.
Despite the impressive progress over the last years, sev-eral different questions related to scrambling and entan-glement propagation still await a more detailed answer. It has been observed that the exponential growth of the
square commutator is connected to the chaotic behaviour of an underlying semi-classical limit. The precise role of semiclassical correlations in determining scrambling dy-namics and its various stages are presently under intense study30–32. Furthermore, in view of the various forms in
which quantum correlations manifests in a many-body system, it is important to understand how entanglement is connected to the scrambling of information. A first connection between square commutators and the spread-ing of quantum entanglement has been made in the con-text of unitary quantum channels33,34. An analysis of
dif-ferent velocities of propagation of information has been performed in [24], while connections of scrambling to the growth of R´enyi entropies and multiple-quantum coher-ence spectra have been investigated in [35–37]. In long-range systems, scrambling has been studied in connection to correlation bounds13,38and its time average as a probe
of criticality39. However, an analysis of the dynamics and
the relevant time-scales in relation to the different pro-cesses involved in the spreading of information is still missing.
In this work, we address all these questions by studying multipartite entanglement propagation and scrambling in spin chains with long-range interaction either subject to a quantum quench or a periodic drive. There are several reasons behind this choice. Spin chains with long-range interactions possess a well defined semiclassical limit, and thus represent a natural playground40 to study the role
of classical correlations in scrambling. Furthermore, they allow exploring the transition from semiclassical to quan-tum dominated regimes in the dynamical behaviour. We will consider both the case of integrable and chaotic dy-namics. Moreover, scrambling is experimentally accessi-ble with long-range quantum simulators, as it has been measured for unitary operators41. We will present results
for the entanglement dynamics of the quantum Fisher in-formation, the tripartite mutual information and opera-tor scrambling, studied via the square commutaopera-tor. As we are going to show in the rest of the paper scrambling
and entanglement dynamics turn out to be very different.
α = 0 Quantum quench Quench at DPT Periodic kicking
Scrambling t < tEhr t2/N3 eλt eλt
tEhr< t < t∗ t4/N4 t/N const.
Entanglement t < tEhr growth peak growth
tEhr< t < t∗ const. const. const.
TABLE I. Scrambling and entanglement dynamics for the different protocols with the infinite range hamiltonian. While for all the dynamics, entanglement grows and saturate at tEhr, scrambling continues his growth in the regular case. Particularly
interesting is that, despite the dynamics being regular, the early-time exponential behaviour emerges when the dynamics occur at the critical point of the dynamical phase transition DPT, see Fig.8. The Ehrenfest time and the recurrence time depend on the dynamics too: tEhr ∝
√
N for the regular quantum quench, tEhr∝ log N for the quench at DPT and for the periodic
kicking, while trec∝ N for the quantum quench dynamics and trec∝ exp(exp(N)) for the chaotic period kicking.
The paper is organized as follows. The next section is devoted to a summary of the results with a direct com-parison between multipartite entanglement growth and scrambling. In sectionIII, we review the long-range ver-sion of the Ising chain and the type of dynamics that are considered across the paper. We recall the semiclassi-cal limit together with the quantum and classisemiclassi-cal char-acterization of chaos. In section IV, we briefly review the definitions of the quantities under consideration: the quantum Fisher information, the tripartite mutual infor-mation, and the square commutator. In section V we describe the different numerical and the semi-analytical methods used to reproduce the behavior of the square-commutator. We first present in sectionVI Athe results for the entanglement dynamics and its semiclassical na-ture for sufficiently long-range interaction. We discuss how this behavior changes when the range of the interac-tion is decreased. Then, in secinterac-tionVI B, we consider the results for the square-commutator and we argue that the long-time dynamics of the square commutator accounts for the quantum chaoticity of the dynamics. We provide evidence for our claims by discussing an example of an exponential growth of scrambling in the case of a reg-ular quantum dynamics. Section VII is devoted to our conclusions.
II. MAIN RESULTS
In this paper, we study how entanglement and operator’s scrambling grow and spread in Ising spin chains with two-body power-law decaying interactions, Jij ∝ |i − j|−α. We consider the case in which an initial separable state, i.e. |ψ0i = | ↑ ↑ . . . ↑ i, is brought out-of-equilibrium by means of a quantum quench or a periodic drive. Our findings can be summarized as follows:
1. Entanglement dynamics reflects the semi-classical nature of the system: it is weak, slowly growing and saturating at the Ehrenfest time tEhr. This is
what lies at the heart of the classical “simulabil-ity” of quantum long-range interacting systems in the context of MPS-TDVP42,43 with small
bond-dimension as well as semiclassical methods44–46.
2. The square commutator is characterized by two dif-ferent regimes, a first semiclassical growth up to the tEhr (exponential for chaotic dynamics), fol-lowed by a fully quantum non-perturbative poly-nomial growth (saturation for chaotic dynamics), symmetric around t∗ = t
rec/2 the recurrence time trec. We show that the initial growth encodes the nature of classical orbits and can be exponential also for regular integrable dynamics, provided they have some classical instabilities. Conversely, the second regime accounts for the quantum chaoticity of the dynamics, see tableI.
3. The dynamics of the information spreading changes with the range of interaction α. The Hamiltonian with 0≤ α < 1 is dominated by the classical limit and the structure of entanglement and scrambling is the same as in the infinite range case, see table
I. For 1 ≤ α < 2, the entanglement grows linearly in time and the structure of the asymptotic state is the same as for α < 1. For α≥ 2 the state displays the typical entanglement dynamics and structure of short-range interacting systems, with negative TMI.
This shows that state’s entanglement growth and op-erator’s scrambling are two distinct, apparently discon-nected phenomena. Interestingly, this becomes glaringly obvious in the regular regime, rather than in the chaotic one, see Fig.4.
3
III. THE MODEL AND OUT-OF-EQUILIBRIUM PROTOCOLS
We consider an Ising chain in transverse field with long-range interactions, ˆ H =−12 N X i6=j Jijˆσizσˆzj− h N X i ˆ σix , (1) where σˆx
i, ˆσiz are spin operators and Jij = J|i − j|−α/N (α) and N (α) = PN
r=11/rα is the Kac normalization47. The (solvable) infinite range limit
α = 0 of Eq.(1) is known as the Lipkin-Meshov-Glick model (LMG)48 and it has been intensively studied
out-of-equilibrium6,49–52. In this case the Hamiltonian
conserves the total spin and we restrict our analysis to the sector of the ground-state S = N/2. This has a semiclassical limit before tEhr, controlled by~eff =~/N, where the system can be described classically in terms of only two degrees of freedom {Q, P } and a classical Hamiltonian6,49,50. For finite N , ground states of the
Hamiltonian of Eq.(1) can be seen as coherent wave-packets with width σ∼√~eff that evolve for short times classically6. The semi-classical dynamics is discussed in
details in next section. As far as the dynamics of local observables is concerned, the Hamiltonian of Eq.(1) is found to behave as the infinite range for α < 1, as a short-range one for α > 27,10.
Taking an initial separable state totally polarised along the z axis|ψ0i = | ↑ ↑ . . . ↑ i, we probe entanglement dy-namics and scrambling with the two following protocols. a. Quantum quench The state|ψ0i is evolved with the Hamiltonian (1) with a transverse field hf. For α ≤ 1, a special case, important also for the present analysis, is represented by hf = hc = 12 where a dynamical phase transition (DPT) occurs53, whose origin can be
traced back to the corresponding classical dynamics. Away from the dynamical critical point the Ehrenfest time reads tr
Ehr ∝ √
N while at the dynamical critical point tc
Ehr ∝ log N. It was shown in [39] that DPT can be detected with the average value of out-of-time correlators.
b. Periodic kicking In order to address chaotic dynam-ics in long-range spin system we will also consider the case in which periodic kicks are added to the evolution governed by Eq.(1), with α = 0. This model, known also as the “kicked top” for h = 0, is a paradigmatic example of the standard quantum chaos54,55. The time-evolution
operator over one period reads ˆ
U = ˆUkexp h
−i ˆH τi with ˆUk≡ exp
−i2 KN Sˆz2
. (2)
Depending on the value of the kicking strength K, this model is known to exhibit a transition between a regular
regime and a chaotic one54,55. When K 1 ∀h
f, orbits deviate exponentially in time and tc
Ehr ∝ log N. A. Semiclassical phase-space
Let us recall the main features of the semiclassical dy-namics. Since the Hamiltonian of Eq.(1) commutes with the total spin ˆS =Pi ˆSi, we restrict ourself to the spin sub-sector of the ground state S = N/2, where the di-mensionality of the Hilbert space is N + 1. Defining
ˆ
m ≡ ˆS/S, we can re-express the LMG Hamiltonian in terms of its components
ˆ HLM G=− N J 2 mˆ z 2 − h ˆmx . (3)
This allows to consider an effective~eff = N~ that iden-tifies the semiclassical limit with the large-N mean field one. In what follows we set~ = 1. In this limit, the sys-tem is effectively described by the classical Hamiltonian
H0(Q, P )≡ − J 2Q
2
− hp1− Q2 cos (2P ) , (4) where the two conjugate variables Q, P are given in terms of the expectation values of ˆm on a wave packet as mz = Q, mx = p1
− Q2cos(2P ) and my=p1
− Q2sin(2P ) and obey to the classical Hamil-ton equations,6,49,50.
In the sudden quench case, the ground state at h0 is evolved with the hamiltonian with transverse field hf. The dynamical phase transition DPT between a finite and zero order parameter occurs at hf = hc= (h0+1)/2. One can define a dynamical order parameter as the average magnetization in time: Q = limT→∞R0TQ(s)ds, which is different from zero in the symmetry broken phase. Indeed, at hc the phase point associated to the initial ground state energy (which is conserved) lays right on the separatrix of the final Hamiltonian: for hf > hc it orbits around the maximum with a Q = 0, while for hf < hc it orbits around one of the two ferromagnetic minima and Q6= 0, see Fig.1.
When the kicking is added, the total classical Hamilto-nian reads
H(Q, P, t) = H0(Q, P ) +Hkick(Q, P ) X
n
δ(t− nτ) , (5)
where Hkick(Q, P ) = −K2Q2: the classical kicking acts every period τ like a rotation around the z axis with an angle proportional to mz. In the numerical calculations, we re-express the Hamilton’s equation of motion as equa-tion of moequa-tions for the spin-components m
˙ mx(t) = 2 J my(t) mz(t) ˙ my(t) = 2h mz(t)− 2 J mx(t) mz(t) ˙ mz(t) =−2h my(t) . (6)
.8 .6 .4 .2 0 .2 .4 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Q(t) .8 .6 .4 .2 0 .2 .4 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Q(t) 0.8 0.6 0.4 0.2 0 0.2 0.4 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 P (t )/ ⇡ Q(t) (a.) (b.) (c.)
FIG. 1. Classical phase space (a.) and Poincar´e sections of the classical limit of the model (b., c.). (a.) Phase space for the hamiltonian of Eq.(4) for hf = 12. In purple the separatrix between orbits with Q = 0 and Q6= 0, which corresponds at the
ground state energy for h0 = 0. (b.) Regular Poincar´e section for K = 0.2: we see that the dynamics remains always regular
and each trajectory is a closed curve. (c.) Chaotic Poincar´e section for K = 20: the system clearly becomes chaotic and the trajectories tend to cover all the phase space. (hf = 2, τ = 1). The red dot in the plots represents the initial condition in the
classical limit, while the orange points represent its stroboscopic evolution. Note that these equations can be obtained also from the
expectation value of the Heisenberg equation of motion, setting to zero the second order cumulant. This is justi-fied by the fact that the magnetization components com-mute in the classical limit: mˆα, ˆmβ= i
N/2αβγmˆ γ. In the limit of large but finite N , one can consider the semiclassical WKB approximation6 and explore
wave-packet dynamics. In this framework, ground states of H0 can be seen as coherent wave packets with width σ = √~eff = 1/√N . This semiclassical picture holds until the states behave like well defined wave-packets. It is then natural to define the time for which semi-classics breaks down- the Ehrenfest time - as the time for which the initially coherent wave-packet is spread and delocal-ized. It is well known that this depends on the nature of the classical dynamics56
tEhr∼ 1 √ ~eff =√N regular 1 2λln 1 ~eff = 1 2λln N chaotic/unstable , (7) where λ > 0 is the Largest Lyapunov exponent of the classical dynamics in the chaotic case.
B. Characterization of chaos
In the quantum realm, an important signature of chaos is provided by the spectral properties of the evolution operator, in our case by the properties of the Floquet spectrum. The distribution of the Floquet level spacings δα ≡ µα+1− µα (the µα are in increasing order), nor-malized by the average density of states, gives informa-tion on the integrability and ergodicity properties of the system55,57–59: if the distribution is Poisson, then the
sys-tem is integrable; if it is Wigner-Dyson, then the syssys-tem is ergodic. In order to probe the integrability/ergodicity
0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0 2 4 6 8 10 12 14 <r> λ
FIG. 2. Regular-chaotic transition witnessed by the average level spacing ratio. (N = 1000, τ = 1, h = 2). Throughout the paper, we always chose K = 20, which clearly displays chaotic dynamics.
properties through the level spacing distribution, we con-sider the so-called level spacing ratio
0≤ rα≡
min{δα, δα+1} max{δα, δα+1} ≤ 1 .
(8) The different level spacing distributions are character-ized by a different value of the average r ≡ hrαi over the distribution. From the results of Ref. [60], we ex-pect r = 0.386 if the system behaves integrably and the distribution is Poisson; on the other side, if the distri-bution is Wigner-Dyson and the system behaves ergod-ically, then r = 0.5295. In our case, the Floquet levels fall in two symmetry classes, according with the corre-sponding Floquet state being an eigenstate of eigenvalue +1 or 1 of the operator eiπ ˆSx under which the
Hamilto-nian is symmetric54. Therfore we need to evaluate the
level spacing distribution and the corresponding r only over Floquet states in one of the symmetry sectors of the
5
Hamiltonian. The level spacing ratio for this model is reported in Fig.2as a function of the kicking strength K: it shows a transition from a regular to a chaotic regime. The relationship between classical chaos and the prop-erties of the many-body quantum dynamics has been widely studied in the past, giving rise to a plethora of signatures of chaos in the quantum domain55,57.
Classically, a system is ergodic if all the trajectories uniformly explore the accessible part of the phase space. In case of few degrees of freedom, a qualitative measure of this phenomenon is the Poincar´e section: some initial values are evolved under the stroboscopic dynamics reporting on a P, Q plot the sequence of their positions. If the initial condition lies in a regular region of the phase space, our points will be over a one-dimensional manifold. If instead, the initial condition is in a chaotic region of the phase space, our points will fill a two-dimensional portion of phase space. The model under analysis satisfies this conditions in the semi-classical limit, see Fig.1.
IV. CHARACTERIZATION OF ENTANGLEMENT AND SCRAMBLING Let us now introduce the quantities that we will use to characterize entanglement and scrambling. As far as the entanglement is concerned, we will focus on the multi-partite case (bimulti-partite entanglement was already studied in [61 and 62]). The characterisation of multipartite entanglement is more delicate than that of bipartite entanglement since there exist a zoo of possible measures and witnesses. We will focus here on the quantum Fisher information (QFI) FQ(t) and on the tripartite mutual information (TMI) I3(t)33, which accounts for the information delocalization. Scrambling is instead studied via the square commutator c(t).
The quantum Fisher information is a witness of mul-tipartite entanglement which has been shown to obey scaling at the equilibrium transition point63and is
con-nected to the diagonal ensemble in the non-equilibrium case 64. The QFI gives a bound on the size of the
biggest entangled block. For example, given a system of N spins, if the QFI density fQ ≡ FQ/N > k, then there are at least k + 1 entangled spins65,66. For pure
states, the QFI is given by an optimization over a generic linear combination of local spin operators of FQ( ˆO, t) = 4 h∆ ˆO2it. Here, we consider collective spin operators ˆO = ˆS = 1
2 P
iσˆi and we maximise over the three directions.
The tripartite mutual information is defined as I3(A : B : C) = I(A : B) + I(A : C)− I(A : BC) where A, B, C, D are four partitions and the quantity I(A : B) is the mutual information between A, B. This takes into account information about A that is non-locally
stored in C and D such that local measurements of B and C alone are not able to re-construct A. Usually I3 < 0 is associated with the delocalisation of quan-tum information in the context of unitary quanquan-tum channels33. In this case, more appropriately, we study
the delocalisation of the initial state information under the dynamics, which is a complementary measure of entanglement.
Finally, in order to characterise the dynamics of scrambling, we will focus on the square commutator c(t) = −h[ ˆB(t), ˆA]2
i. This object measures the non-commutativity induced by the dynamics between two ini-tially commuting operators ˆA and ˆB. It was introduced by Larkin and Ovchinnikov in [20], to describe semiclassi-cally the exponential sensitivity to initial conditions and the associated Lyapunov exponent. By taking collective spin operators29A = ˆˆ B = ˆm
z = ˆSz/S, the square com-mutator has a natural classical limit for~eff→ 067
c(t) =− h [ ˆmz(t), ˆmz]2i → ~2
eff{Q(t), Q(0)}2, (9) where Q(t) = h ˆmz(t)i on a coherent wave packet, {·} are the Poisson brackets of the corresponding classical trajectory and the average (·) is performed over an initial phase-space distribution.
V. METHODS
The results presented in this work were obtained with a series of numerical techniques and two semi-analytical approximations.
The numerical methods are a combination of exact diag-onalization (ED) and well-established semi-classical ap-proximations which are based on Wigner phase-space representations: the truncated Wigner approximation (TWA)68on the continuum phase-space and the discrete
truncated Wigner approximation (DTWA)45,69of the
fi-nite dimensional phase space. To this end, we gener-alised the corresponding expression for the square com-mutator to the discrete phase space representation, see Eq.(20) and the supplementary material for the details used in our calculations. All these approaches neglect terms of the order of O(1/N) and give the same re-sults up to the Ehrenfest time. DTWA, in particular, is able to reproduce also entanglement long-time dynamics. Furthermore, we also adopted the matrix product state time-dependent variational principle (MPS-TDVP)42,43,
for the dynamics of long-range hamiltonians with α6= 0. We combine these approaches with two semi-analytical methods in order to predict the behavior of c(t) up to tEhr. The first method is a equation of motion closure at fifth order : it consists in deriving a hierarchy of differen-tial equations for the square commutator and in closing it by setting the fifth order cumulant to zero. This allows to decouple the higher order commutator and to close the system of equations. By setting the appropriate initial
0 10 20 30 40 50 0 2 4 6 8 10 c( t) t J ED Cumulant closure solution Gaussian spin-waves TWA DTWA
FIG. 1. Comparison between the di↵erent semi-classical ap-proximations of the square commutator. Quantum quench dynamics with hf = 2 and N = 50. All the approximated
methods are found to reproduce the dynamics of c(t) up to times/pN , indicated with a thin line in the plot.
cated Wigner Approximation on the continuum phase-space and the Discrete Truncated Wigner Approxima-tion of the finite dimensional phase space. For a survey on Wigner representations see [5, 6] on the continuum phase-space and [7, 8] for its discrete version.
These four methods turn out to be equivalent for the scrambling dynamics and they give the results of Fig.1. In the following paragraphs we give a survey of all these numerical methods.
Fifth order cumulant approximation The cumu-lant closure at order n is a general method that consists in closing a set of di↵erential equations, by setting to zero all cumulants of the order n. A very easy example is the cumulant closure at second order, which allows to compute the classical equation of motion for the magne-tization of Eq.(4). We are interested in the dynamics of the square commutator (see Eq.(3) of the main text) and we wish to find a set of di↵erential equations that gives its evolution.
One first defines a symmetric (n + m + 2)-string commu-tator c↵1,...↵n, 1... m(t) = 1 2h [ ˆm ↵1(t) ˆm↵2(t) . . . ˆm↵n(t), ˆmz] ⇥ ˆ m 1(t) ˆm 2(t) . . . ˆm m(t), ˆmz⇤ i 1 2h ⇥ ˆ m 1(t) ˆm 2(t) . . . ˆm m(t), ˆmz⇤ [ ˆm↵1(t) ˆm↵2(t) . . . ˆm↵n(t), ˆmz]i , (6) where there are n + m time-dependent operators and two time-independent ones. Within this notation, the square commutator of Eq.(1) reads c(t) = cz,z(t). The dynam-ics will generate an infinite number of coupled equations
of motion; this hierarchy of di↵erential equations can be closed by setting the fifth order cumulant to zero: hABCDEic = 0. If one assumes that the magnetiza-tion is classical (second order cumulant set to zero), the (3 + 2)-string commutator decouples in
c↵, (t) + c↵, (t) = 2⇥m (t) c↵, (t) + m (t) c↵, (t)⇤ , (7) for ↵, , 2 {x, y, z}. This allows to close the hierarchy of di↵erential equations, which are coupled to the classi-cal magnetization dynamics of Eq.(4) as
8 > > > > > > > > > > > < > > > > > > > > > > > : ˙cz,z = 4 h cz,y ˙cz,y= 2 h cy,y+ 2 h cz,z 2 J [cz,zmz+ cz,xmx] ˙cy,y= 4 h cy,z 4 J [cx,ymz+ cx,ymz]
˙cx,y = 2 h cx,z 2 J [cx,xmz+ cx,zmx] +2 J [cy,ymz+ cy,zmy]
˙cx,z = 2 h cx,y+ 2 J [cz,zmy+ cz,ymz] ˙cx,x= 2 J [cx,zmy+ cx,ymz] .
(8) These equations are integrated numerically via the fourth order Runge-Kutta method. The information about the initial state and the dimension of the system is encoded in the initial conditions. The cz,z(t) that we obtain with this cumulant closure give the limit N ! 1 of all these semiclassical approximations. It well reproduces the ex-act c(t), up to a time tr
Ehr (see Fig.1).
Spin wave approximation on top of mean field Quantum fluctuations are treated as small fluctuations on top of the classical solution [9]. One first produces a time-dependent rotation of the reference frame R = ( ˆX(t), ˆY (t), ˆZ(t)), in such a way that the ˆZ(t) axis fol-lows the motion of the classical collective spin hˆS(t)i. Then an Holstein-Primako↵ transformation is performed and the quantum fluctuations are kept at the gaussian order. In this rotating frame the collective spin opera-tors are the zero-mode components in the Fourier Trans-form: ˜0
a, with a 2 R. Our approximation consists in taking the operator on the Z-axis not varying in time: ˜0
Z(0) ⇠ ˜Z0(t) +O((N/2) 1). This allows to compute commutators in the rotating frame, hence to get an ap-proximated solution for c(t).
The main steps to solve the dynamics are the following [9]:
• perform a time dependent unitary rotation with V ( ✓(t), (t) ) = e i (t)2 P iˆ z ie i ✓(t) 2 P iˆ y i; the
an-gles are chosen such that h ˆSXi = h ˆSYi = 0. In the new frame the operators evolve with ˜H = V H V†+ iV ˙V†;
• perform an Holstein-Primako↵ transformation on the operators in R in terms of the conjugate vari-ables (˜q0, ˜p0);
FIG. 3. Comparison between the different semi-classical ap-proximations of the square commutator. Quantum quench dynamics with hf = 2 and N = 50. All the approximated
methods are found to reproduce the dynamics of c(t) up to times tr
Ehr, indicated with a thin line in the plot.
conditions, one can integrate numerically the equations and get the approximated c(t). The second method is a time-dependent Holstein Primakoff and it consists in including quantum fluctuations on top of the classical result and to keep it only at the Gaussian level. These approaches turn out to be equivalent and to correctly re-produce c(t) before tEhr as in Fig.3. The following two paragraphs are devoted to a description of these approx-imations.
Equation of motion closure at fifth order The cumulant closure at order n is a general method that consists in closing a set of differential equations, by setting to zero all cumulants of the order ≥ n. A very easy example is the cumulant closure at second order, which allows computing the classical equation of motion for the magnetization of Eq.(6). We are interested in the dynamics of the square commutator and we wish to find a set of differential equations that gives its evolution. One first defines a symmetric (n + m + 2)-string commu-tator cα1,...αn,β1...βm(t) =− 1 2h [ ˆm α1(t) ˆmα2(t) . . . ˆmαn(t), ˆmz]× ˆ mβ1(t) ˆmβ2(t) . . . ˆmβm(t), ˆmz i −12hmˆβ1(t) ˆmβ2(t) . . . ˆmβm(t), ˆmz × [ ˆmα1(t) ˆmα2(t) . . . ˆmαn(t), ˆmz]i , (10) where there are n + m time-dependent operators and two time-independent ones. Within this notation, the square commutator of Eq.(1) reads c(t) = cz,z(t). The dynamics will generate an infinite number of coupled
equations of motion. We close this hierarchy of differ-ential equations by setting the fifth order cumulant to zero: hABCDEic = 0. If one assumes that the magne-tization is classical (second order cumulant set to zero), the (3 + 2)-string commutator decouples in
cα,βγ(t) + cα,γβ(t) = 2mβ(t) cα,γ(t) + mγ(t) cα,β(t) , (11) for α, β, γ∈ {x, y, z}. This allows to close the hierarchy of differential equations, which are coupled to the classi-cal magnetization dynamics of Eq.(6) as
˙cz,z =−4 h cz,y ˙cz,y=−2 h cy,y+ 2 h cz,z− 2 J [cz,zmz+ cz,xmx] ˙cy,y= 4 h cy,z− 4 J [cx,ymz+ cx,ymz]
˙cx,y = 2 h cx,z− 2 J [cx,xmz+ cx,zmx] +2 J [cy,ymz+ cy,zmy]
˙cx,z=−2 h cx,y+ 2 J [cz,zmy+ cz,ymz] ˙cx,x= 2 J [cx,zmy+ cx,ymz] .
(12) These equations are integrated numerically via a fourth order Runge-Kutta method. The information about the initial state and the dimension of the system is encoded in the initial conditions. The cz,z(t) that we obtain with this cumulant closure turns out to reproduce the limit N→ ∞ of all these semiclassical approximations. It well reproduces the exact c(t), up to a time tr
Ehr (see Fig.3). Time-dependent Holstein Primakoff In a spin wave expansion, quantum fluctuations are treated as small fluctuations on top of the classical solution70. One
first produces a time-dependent rotation of the reference frame R = ( ˆX(t), ˆY (t), ˆZ(t)), in such a way that the
ˆ
Z(t) axis follows the motion of the classical collective spinhˆS(t)i. Then an Holstein-Primakoff transformation is performed and the quantum fluctuations are kept at the gaussian order. In this rotating frame the collec-tive spin operators are the zero-mode components in the Fourier transform: ˜σ0
a, with a ∈ R. Our approximation consists in taking the operator on the Z-axis not vary-ing in time: ˜σ0
Z(0) ∼ ˜σZ0(t) +O((N/2)−1). This allows to compute commutators in the rotating frame, hence to get an approximated solution for c(t).
The main steps to solve the dynamics are the following70:
• perform a time dependent unitary rotation with V ( θ(t), φ(t) ) = e−iφ(t)2 P iσˆ z ie−i θ(t) 2 P iσˆ y i; the
an-gles are chosen such that h ˆSXi = h ˆSYi = 0. In the new frame the operators evolve with ˜H = V H V†+ iV ˙V†;
• perform an Holstein-Primakoff transformation on the operators in R in terms of the conjugate vari-ables (˜q0, ˜p0);
• keep only Gaussian terms, which is equivalent to neglect allO((N/2)−3/2) terms in the equations.
7
With such a choice one remains with the following hamil-tonian ˜ H N = hclass(t)+ 1 √ N shlin(t)+ 1 N shquad(t)+O((Ns) −3/2) , (13) Then, by settingh ˆSXi = h ˆSYi = 0, one gets the equation of motion for the rotating frame, see [71]
(
˙θ = 2J sin θ cos φ sin φ ˙
φ =−2h + 2J cos θ cos2φ . (14) In the same way one can obtain the Heisenberg equation of motion for for ˜q0, ˜p0. Further defining the zero-mode fluctuations as ∆qq0 (t)≡ h ˜q0(t) ˜q0(t)i (15a) ∆pp0 (t)≡ h ˜p0(t) ˜p0(t)i (15b) ∆qp0 (t)≡ 1 2h ˜q0(t) ˜p0(t) + ˜p0(t) ˜q0(t)i . (15c) and combining them with the equations for ˜q0, ˜p0, one gets the equations of motion for the zero-mode fluctua-tions ˙
∆qq0 = 4J cos θ sin φ cos φ ∆ qq
0 + 4J cos2φ− sin2φ
∆pq0 ˙
∆pp0 =−4J cos θ sin φ cos φ ∆ pp 0 − 4J cos2φ sin2θ ∆ pq 0 ˙ ∆pq0 =−2J cos2φ sin2θ ∆ qq 0 + 2J cos2φ− sin2φ ∆pp0 (16) They are a set of linear time-dependent differential equa-tions, which can be solved numerically with the appro-priate initial conditions. They are exactly the quantities that appear in the computation of the square commu-tator. In order to compute it perform first a rotation ˜
σα
0(t) to ˜σa0(t) with V (θ(t), φ(t)), then compute commu-tators like [˜σa
0(t), ˜σ0Z], noticing that ˜σZ0(0) = ˜σZ0(t) + O((Ns)−1), hence at this order they are equal-time com-mutators that give rise to the zero mode fluctuations of Eq.(15). For example our square commutator of Eq.(3) reads as
c(t) = sin φ2∆pp0 +cos2θ cos2φ ∆ qq
0 −2 cos θ sin φ cos φ ∆ pq 0 , (17) which can be obtained numerically from the integration of Eq.(16) and gives exactly the same result of the pre-vious approximation, see Fig.3. This is correct until the spin-wave density remains small, which, for finite N , oc-curs before tEhr.
Notice that this method could be in principle extended to long range systems with α6= 0 and other variations of fully connected models70. In addition one could in prin-ciple go beyond the gaussian approximation by keeping the interaction between spin waves.
VI. RESULTS
As we hinted out at the beginning of the paper, entangle-ment and scrambling are two different phenomena, char-acterized by different time scales, see Fig.4. Let us now
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 t/N
Tripartite information with nA= 1 nB= 10and nC= 20, hf= 2
N = 50 N = 100 N = 200
Figure 31: From h0= 0to hf= 2. Tripartite mutual information for long times
0 0.5 1 1.5 2 2.5 3 3.5 0 50 100 150 200 t J Information dynamics fQ(t)/N I3 c(t)
Figure 32: System with N = 100. Dynamics from h0= 0to hf = 2. Tripartite information with nA= 1 nB= 10and nC= 20.
6.3 Square commutator with hf> 1
19
FIG. 4. Quantum information’s dynamics for the regular dy-namics. The entanglement quantities, QFI and TMI (red and yellow), saturate at tEhr, while the square commutator of the
longitudinal magnetization operator (blue) goes beyond semi-classics and keeps growing up to t∗. Exact diagonalization results for N = 100, hf = 2, TMI with nA= 1 nB = 10 and
nC= 20.
finally describe in details the results obtained for the dy-namics of entanglement and scrambling using the meth-ods described before.
A. Entanglement dynamics
In the infinite range model, entanglement dynamics and information delocalisation reflect the semiclassical nature of the system under analysis. We start discussing the dynamics governed by Eq.(1) after a quantum quench and describe afterward the case of the periodic kicking protocol.
Let us focus first on the LMG model at α = 0. Both fQ(t) and I3(t) have the same dynamics; growth followed by saturation at tEhr, as dictated by the semiclassical dy-namics of the model, see Fig.5 (top and middle panels). The stationary state displays global entanglement of gen-uine multipartite nature fQ= φQN , where φQ≤ 1/2 is a function of the transverse field. The value of the phase φQ along the z direction can be computed analytically in terms of elliptic integrals. Following Ref. [71], with a combination of the classical equation of motion and energy conservation, defining k = J/2h≥ 1, one gets
φzQ = 1 k2 " (k2− 1) +E(θF (θk, k) k, k)− π 2F (θk, k) 2# , (18) where F (φ, k), E(φ, k) are the elliptic integrals of first and second kind of amplitude φ, modulus k and θk = arcsin(1/k) is the inversion point of the classical trajec-tory Q(t). The maximum asymptotic entanglement wit-nessed by the QFI is fQ= N2, which occurs when the
sys-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 fQ (t )/N h < hc h = hc h > hc I3 (t ) 0 0.25 0.5 0.75 0 50 100 150 200 250 300 350 400 Exact DTWA t J fQ (t )/N
FIG. 1. QFI and TMI (top and middle) during the evolution after a quantum quench, performed below, at and above the DPT. Entanglement grows in time up to tr
Ehr for quenches
above and below the dynamical phase transition (green and blue) and at tcEhr at the critical point. Long-time dynamics
(bottom) of the QFI, compared with the semiclassical approx-imation. The DTWA is able to reproduce the dynamics up to tr
rec and beyond. (Top and middle) N = 450, hf = 0.2,
hf = 0.5 and hf = 2; for the I3: nA= 1, nB = 50, nC= 200.
(Bottom) N = 100 and hf = 2, DTWA obtained with 5· 103
samplings.
regime using DTWA, spin-wave theory and cumulant clo-sure methods [48]. All these approaches neglect correc-tionsO(1/N) and give the same results before the Ehren-fest time. The accuracy of all the semiclassical analysis is justified by the entanglement structure itself. In fact this is what lies at the heart of the classical “simulability” of quantum long-range interacting systems in the context of MPS-TDVP [59, 60] and with semiclassical methods [58, 61]. DTWA in particular is able to reproduce also the long-time dynamics even beyond the recurrence time tr
rec/ N as shown in Fig.1 (bottom panel). This is due to the discrete nature of the method that mimics the discreteness of the spectrum, responsible for the recur-rences [58]. The same asymptotic structure and dynam-ics is found for all mean-field like systems 0 ↵ < 1: the QFI grows linearly in time up to a value⇠ N, and the TMI increases logarithmically in time up to a con-stant value. For 1 ↵ < 2, the QFI and the TMI grow linearly in time and the entanglement structure of the asymptotic state is the same as for ↵ < 1. Decreasing the range of interaction the situation changes: for ↵ 2 the state displays the typical dynamics and structure of
0 0.1 0.2 0.3 0.4 0.5 (a.) 0 0.01 0.02 0.03 (b.) 0 0.1 0.2 0.3 (c.) 0 0.1 0.2 0.3 (d.) 0.9 0.6 0.3 0 0.3 0 1 2 3 4 5 6 (e.) 0 10 20 30 40 50 0.02 0 0.02 0.04 (f.) 0 ↵ < 1 hf= 1.5 hf = 0.25 1 < ↵ < 2 ↵ > 2 t J t J
FIG. 2. We plot the minimal TMI (dotted lines) and the fQ(t)/N (full lines) as a function of time, above (left panels)
and below the DPT (right panels). We obtained the minimal TMI by calculating the tripartite mutual information for all possible partitions A,B,C,D of the system and then taking the minimum. For ↵ < 1 (a., b.) the dynamics is the same of the LMG model. When ↵ > 2 (e., f.), the QFI does not scale with the system size and remains bounded with time, whereas the minimum of the TMI decreases linearly with time and becomes negative for longer times. In all other cases we observe linear growth of the QFI with time, whereas the min-imum of the tripartite mutual information remains bounded with time. We present the detailed scaling with the system size in the supplementary material. The parameters of the evolution are: ↵ = 0.5, ↵ = 1.5, ↵ = 2.5, hf = 0.25 and
hf = 1.5. System sizes N = 200 (or N = 100 for ↵ = 2.5).
short range interacting systems fQ⇠ const; interestingly I3 < 0 signaling that the information about the initial condition is spread throughout the degrees of freedom of the state (see Fig.2).
We conclude the analysis of the multipartite entangle-ment by considering the kicked case in the regime when dynamics is chaotic. This system heats up to a state where all local observables on any Floquet state corre-spond to the infinite temperature values [46]. The en-tanglement saturates at tc
Ehr to the corresponding val-ues, fQ = 1 +N3 +O(1/N) and I3= log[nAnBnC(nA+ nB+ nC)]/[(nA+ nB)(nA+ nC)(nB+ nC)] (see [48]). Square commutator — Scrambling, as measured by he square commutator, behaves in a way profoundly dif-ferent from the entanglement. It is characterised by a first semi-classical regime and a second quantum non-perturbative growth. Interestingly, this phenomenon is very evident in the regular regime Fig.(3), and it is much less clear in the chaotic one Fig.(4). In the case of the FIG. 5. QFI and TMI (top and middle) during the evolution
after a quantum quench, performed below, at and above the DPT. Entanglement grows in time up to tr
Ehr for quenches
above and below the dynamical phase transition (green and blue) and at tcEhr at the critical point. Long-time dynamics
(bottom) of the QFI, compared with the semiclassical approx-imation. The DTWA is able to reproduce the dynamics up to trrec and beyond. (Top and middle) N = 450, hf = 0.2,
hf = 0.5 and hf = 2; for the I3: nA= 1, nB= 50, nC= 200.
(Bottom) N = 100 and hf = 2, DTWA obtained with 5· 103
samplings.
tem from a product state is quenched to the maximally paramagnetic phase and corresponds to the biggest fluc-tuations of the collective spin operators.
The TMI gives complementary information: being posi-tive, I3 > 0 it shows that the information of the initial state is not delocalised across the system. Interestingly, by increasing α the TMI becomes negative I3 < 0, see Fig.6.
Let us spend a few words for the quench to the DPT, which occurs at hc = 1/2, see Sec.III. In this case, the en-tanglement dynamics is qualitatively different. QFI and TMI at short times peak at tc
Ehr. After a transient they reach their stationary value, which keeps oscillating with-out recurrences, see Fig.5. This behavior is tightly linked to the existence itself of the DPT, that corresponds to a classical separatrix in phase space: the effective classical trajectory takes time of the order of log N to depart from its initial value. After that, the classical picture is lost and the state is spread over the basis giving a constant entanglement, see Fig.S3.
The entanglement dynamics is reproduced, up to very 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 fQ (t )/N h < hc h = hc h > hc I3 (t ) 0 0.25 0.5 0.75 0 50 100 150 200 250 300 350 400 Exact DTWA t J fQ (t )/N
FIG. 1. QFI and TMI (top and middle) during the evolution after a quantum quench, performed below, at and above the DPT. Entanglement grows in time up to tr
Ehr for quenches
above and below the dynamical phase transition (green and blue) and at tcEhr at the critical point. Long-time dynamics
(bottom) of the QFI, compared with the semiclassical approx-imation. The DTWA is able to reproduce the dynamics up to trrec and beyond. (Top and middle) N = 450, hf = 0.2,
hf = 0.5 and hf = 2; for the I3: nA= 1, nB= 50, nC= 200.
(Bottom) N = 100 and hf = 2, DTWA obtained with 5· 103
samplings.
regime using DTWA, spin-wave theory and cumulant clo-sure methods [48]. All these approaches neglect correc-tionsO(1/N) and give the same results before the Ehren-fest time. The accuracy of all the semiclassical analysis is justified by the entanglement structure itself. In fact this is what lies at the heart of the classical “simulability” of quantum long-range interacting systems in the context of MPS-TDVP [59, 60] and with semiclassical methods [58, 61]. DTWA in particular is able to reproduce also the long-time dynamics even beyond the recurrence time tr
rec/ N as shown in Fig.1 (bottom panel). This is due to the discrete nature of the method that mimics the discreteness of the spectrum, responsible for the recur-rences [58]. The same asymptotic structure and dynam-ics is found for all mean-field like systems 0 ↵ < 1: the QFI grows linearly in time up to a value ⇠ N, and the TMI increases logarithmically in time up to a con-stant value. For 1 ↵ < 2, the QFI and the TMI grow linearly in time and the entanglement structure of the asymptotic state is the same as for ↵ < 1. Decreasing the range of interaction the situation changes: for ↵ 2 the state displays the typical dynamics and structure of
t J 0 0.1 0.2 0.3 0.4 0.5 (a.) 0 ↵ < 1 0 0.1 0.2 0.3 (b.) 1 < ↵ < 2 0.8 0.5 0.2 0.1 0 1 2 3 4 5 6 7 (c.) ↵ 2
FIG. 2. We plot the minimal TMI (dotted lines) and the fQ(t)/N (full lines) as a function of time, above (left panels)
and below the DPT (right panels). We obtained the minimal TMI by calculating the tripartite mutual information for all possible partitions A,B,C,D of the system and then taking the minimum. For ↵ < 1 (a., b.) the dynamics is the same of the LMG model. When ↵ > 2 (e., f.), the QFI does not scale with the system size and remains bounded with time, whereas the minimum of the TMI decreases linearly with time and becomes negative for longer times. In all other cases we observe linear growth of the QFI with time, whereas the min-imum of the tripartite mutual information remains bounded with time. We present the detailed scaling with the system size in the supplementary material. The parameters of the evolution are: ↵ = 0.5, ↵ = 1.5, ↵ = 2.5, hf = 0.25 and
hf = 1.5. System sizes N = 200 (or N = 100 for ↵ = 2.5).
short range interacting systems fQ ⇠ const; interestingly I3 < 0 signaling that the information about the initial condition is spread throughout the degrees of freedom of the state (see Fig.2).
We conclude the analysis of the multipartite entangle-ment by considering the kicked case in the regime when dynamics is chaotic. This system heats up to a state where all local observables on any Floquet state corre-spond to the infinite temperature values [46]. The en-tanglement saturates at tcEhr to the corresponding val-ues, fQ= 1 + N3 +O(1/N) and I3= log[nAnBnC(nA+ nB+ nC)]/[(nA+ nB)(nA+ nC)(nB+ nC)] (see [48]). Square commutator — Scrambling, as measured by he square commutator, behaves in a way profoundly dif-ferent from the entanglement. It is characterised by a first semi-classical regime and a second quantum non-perturbative growth. Interestingly, this phenomenon is very evident in the regular regime Fig.(3), and it is much less clear in the chaotic one Fig.(4). In the case of the FIG. 6. We plot the minimal TMI (dotted lines) and the fQ(t)/N (full lines) as a function of time, varying the range of
interaction α. We obtained the minimal TMI by calculating the tripartite mutual information for all possible partitions A,B,C,D of the system and then taking the minimum. For 0≤ α < 1 (a.) the dynamics is the same as the LMG model. For 1≤ α < 2 (b.) we observe linear growth of the QFI with time, whereas the minimum of the tripartite mutual information remains bounded with time. When α > 2 (c.), the QFI does not scale with the system size and remains bounded, whereas the minimum of the TMI decreases linearly with time and becomes negative for longer times. We present the detailed scaling with the system size in the supplementary material. The parameters of the evolution are: α = 0.5, α = 1.5, α = 2.5, hf = 0.75. Data obtained with TDVP for system sizes
N = 200 and bond dimension D = 256 (or N = 100, D = 512 for α = 2.5).
long times, by a semiclassical approach. We studied this regime using DTWA, spin-wave theory and cumulant clo-sure methods, see Sec.V. All these approaches neglect terms of the order ofO(1/N) and give the same results up to the Ehrenfest time. The accuracy of all the semi-classical analysis is justified by the entanglement struc-ture itself. In fact, this is what lies at the heart of the classical “simulability” of quantum long-range interact-ing systems in the context of MPS-TDVP42,43 and with
semiclassical methods44–46. DTWA, in particular, is able
to reproduce also the long-time dynamics even beyond the recurrence time tr
rec∝ N as shown in Fig.5 (bottom panel). This is due to the fact that method averages over an extensive number of trajectories, hence mimicking a discreteness of the spectrum, responsible for the recur-rences45.
The same asymptotic structure and dynamics is found for all mean-field like systems 0≤ α < 1: the QFI grows linearly in time up to a value ∼ N, and the TMI
in-9 44 10 2 100 102 104 106 108 0.1 1 10 10 5 10 4 10 3 10 2 10 1 100 0.1 1 c( t) N 2 tJ/pN quantum t4 exact DTWA classical t2
FIG. 2. Two times regime of the square commutator in the LGM model. In blue, the c(t) obtained with ED for N = 20, 100, 200, 300, 400 increasing the color’s darkness. The two straight lines are the polynomial fit of the two regimes, they clearly cross at t = tr
Ehr = p
N . In the insert we compare the exact c(t) for N = 20 with the semiclassical approaches: also DTWA fails in reproducing the second quantum regime of the square commutator. TWA and DTWA obtained with Nrandom= 5 103samplings. representation: c(t)! ~ 2 N4 X i,j,k,m x i(0) z j(t) y i(0) y k(0) z m(t) x k(0) 2 , (4) where now ix,y,z are the Weyl transform of the spin op-erators and the average · is computed over the initial discrete Wigner distribution, see the Supplementary Ma-terials for the derivation. At tr
Ehr the quantum regime starts, characterized by a polynomial growth ⇠ (t/N)4 up to a maximum c(tr
⇤) = 2, with tr⇤ = trrec/2. Notice that at this time the square commutator, which goes to zero in the previous regime, is constant and indepen-dent of the system size. The quantum nature of chaos is indeed inscribed in this second part of the dynamics, being recurrence times purely quantum and intimately connected to the integrability properties of the spectrum. Even DTWA, that perfectly gets multipartite entangle-ment dynamics up to tr
rec(see Fig.1) is not able to repro-duce the long time dynamics of the square commutator, see Fig.2. DTWA, despite it keeps N discrete trajecto-ries, makes an important approximation: time dependent
operators are factorized on each site at any time [55]. In
our understanding, this is the main reason of the failure of semi-classics and the key characteristic of the quantum
polynomial regime. In fact, after tEhr, the operator’s
ex-pansion starts developing longer and longer strings and the square commutator re-sums all the correlations, until
t⇤, which corresponds to the time at which the string of
length N occurs [48]. Furthermore, since the operator
10 8 10 6 10 4 10 2 (a.) 0 0.3 0.6 0.9 (b.) 10 8 10 6 10 4 10 2 100 102 0 1 2 3 4 5 6 (c.) 0 0.2 0.4 0.6 0.8 1 1.2 (d.) TWA ED N = 400N = 800 t J TWA ED t J/N N = 800
FIG. 3. Square commutator dynamics after a quantum quench to the DPT point K = 0, hf = 1/2 (a.,b.) and after a periodic kicking K = 20, hf = 2, ⌧ = 0.6 (c., d.). At early times (a.,c.) , they are both characterized by an exponential growth up to tEhr log N , see the right side with the log-scale on the y axis. This regime can be perfectly reproduced by the TWA obtained with Nrandom = 104 samplings. (b., d.) At long times the behavior is di↵erent depending on the quantum integrability properties. (c.) the time is rescaled by N in order to show that c(t) up to a maximum at t⇤ N . (d.) In the Kicked top the c(t) stays constant also at very long times.
is expected to go back to himself at trec, we conjecture
that c(t) is always maximum at t⇤ = trec/2, also for
short-range interacting systems. Quenches to hf ⌧ hc
are characterized by the same time-scales: tr Ehr /
p N and tr
⇤ / N. Anyhow, the dynamics has two di↵erent power law: a semi-classical regime⇠ t/N3 followed by a quantum one⇠ t3/N4. Notice that c(tr
⇤)⇠ 10 3/N and it goes to zero at all times in the thermodynamic limit. Since c(t) amounts for the non-commutativity of ˆmz(t) with the Hamiltonian (1), the operator scrambling is in-deed smaller for small hf. At hf = hc, despite the quan-tum system is integrable, we find that the square com-mutator grows exponentially in time up to tc
Ehr / log N as:
c(t) = e 2t
N3 . (5)
This is due to the existence of the unstable trajectory in the classical dynamics. The exponent in Eq.(5) is twice the eigenvalue of the instability matrix of the separatrix trajectory hc = 2
p
hc(1 hc) = 1 for hc= 1/2. This is valid in general for all the classical trajectories associated with DPT. To our knowledge it is the only example of an early time exponential growth in a many-body regular system. Anyhow, after tcEhr, c(t) keeps growing linearly in time up to the tr⇤ and than it goes back, see Fig.3. We now consider the period kicking, where the dynamics is chaotic. As expected c(t) is initially dominated by the FIG. 3. The two-times regime of the square commutator in
the LMG model after a quantum quench with hf = 2. In the main plot the quantum regime of c(t) for di↵erent N : this regime starts atpN and then c(t) grows polynomially as t4. ED results in blue for N = 20, 100, 200, 300, 400 (increasing color darkness), dashed yellow line for the polynomial fit. In the insert we show the semiclassical regime, comparing the exact c(t) with DTWA, which predicts the ⇠ t2 power-law growth, dashed in the plot. Here N = 20 and DTWA obtained with 5· 103 samplings.
quench dynamics, for hf hc the square commuta-tor is by terised a first semiclassical quadratic growth c(t) / t2/N3 until tr
Ehr. In this regime, semiclassical approximations describes very well the evolution of c(t) and we chose to employ DTWA. To this end we have to generalise the corresponding expression for the square commutator to the discrete phase space representation
c(t)⇠ ~ 2 N4 X i,j,k,m ⇥ xzy ij (t) yzx ij (t) ⇤ [ xzykm(t) kmyzx(t)] (4) where ij↵ (t) = ↵ i(0) @ j(t) @ i(0), with x,y,z
i the Weyl trans-form of the spin operators and the average (·) is com-puted over the initial discrete Wigner distribution [48]. At tr
Ehr the quantum regime starts, characterised by a polynomial growth⇠ (t/N)4up to a maximum c(tr
⇤)⇠ 2. At this time, the square commutator is independent of the system size. Even DTWA, that perfectly gets mul-tipartite entanglement dynamics up to tr
rec (see Fig.1) is not able to reproduce the long time dynamics of the square commutator, see Fig.3. Indeed DTWA, despite keeping N discrete trajectories, represents all operators as factorised on each site at any time [58]. At times longer than tEhr, the operator expansion starts develop-ing longer and longer strdevelop-ings and the square commutator re-sums all the correlations, until tr
⇤, which corresponds to the time at which the string of length N occurs [48]. Quenches to hf ⌧ hcare characterized by the same time-scales tr
Ehr and tr⇤ and the same semi-classical regime 10 8 10 6 10 4 10 2 (a.) 0 0.3 0.6 0.9 (b.) 10 8 10 6 10 4 10 2 100 102 0 1 2 3 4 5 (c.) 0 0.2 0.4 0.6 0.8 1 1.2 (d.) TWA ED N = 400N = 800 t J TWA ED t J/N N = 800
FIG. 4. Square commutator dynamics (a., b.) for the LMG model past a quantum quench to the DPT point and (c., d.) for the kicked top with K = 20, hf = 2, ⌧ = 0.6. At early times (a., c.), they are both characterized by an exponential growth up to tcEhr, see the right side with the log-scale on the y axis. This regime can be perfectly reproduced by the TWA. (b., d.) At long times the behaviour is di↵erent depending on the quantum integrability properties. In (c.) the time is rescaled by N in order to show that c(t) grows up to a maximum at tr⇤. In (d.) In the Kicked top the c(t) stays constant also at very long times. In (a., c.) we used N = 800 and TWA obtained with 104samplings
⇠ t2/N3 up to tr
Ehr. Anyhow, the result at long-times is qualitatively di↵erent: the quantum regime is ⇠ t3/N4 and c(tr
⇤)⇠ 10 3/N goes to zero at all times in the ther-modynamic limit. This is a direct consequence of the existence of the dynamical transition, which is detected by the scrambling [38]. Due to the presence of a macro-scopic magnetization, the support of the operators has a constrained dynamics and it will not acquire a string of length N . A special case is represented by the quench at hf = hc; despite the quantum system is integrable, we find that the square commutator grows exponentially in time up to tc
Ehras: c(t) = e2t/N3. This is due to the exis-tence of the unstable trajectory in the classical dynamics. The exponent is twice the eigenvalue of the instability matrix of the separatrix trajectory hc = 2
p
hc(1 hc) for hc= 1/2. This is valid in general for all the classical trajectories associated with DPT. To our knowledge it is the only example of an early time exponential growth in a many-body regular system. Anyhow, after tc
Ehr, c(t) keeps growing linearly in time up to the tr
⇤ and then it goes back, see Fig.4. Long-range interactions do not change drastically this analysis. In the range ↵ < 1, the early time dynamics is the same of what described be-fore. The square commutator grows like a power law at small times, even for ↵ > 2.
We conclude by considering the kicking, which induces a chaotic dynamics. As expected c(t) is initially dom-inated by the classical exponential growth, then as
FIG. 7. The two-times regime of the square commutator in the LMG model after a quantum quench with hf = 2. In the
main plot the quantum regime of c(t) for different N : this regime starts at√N and then c(t) grows polynomially as t4. ED results in blue for N = 20, 100, 200, 300, 400 (increasing color darkness), dashed yellow line for the polynomial fit. In the insert we show the semiclassical regime, comparing the exact c(t) with DTWA, which predicts the ∼ t2 power-law
growth, dashed in the plot. Here N = 20 and DTWA obtained with 5 · 103samplings.
creases logarithmically in time up to a constant value. For 1 ≤ α < 2, the QFI and the TMI grow linearly in time and the entanglement structure of the asymptotic state is the same as for α < 1. Decreasing the range of interaction the situation changes drastically: for α ≥ 2 the state displays the typical dynamics and structure of short-range interacting systems fQ ∼ const; interestingly I3 < 0 signaling that the information about the initial condition is spread throughout the degrees of freedom of the state (see Fig.6). The results are obtained with TDVP, see the supplementary material for a discussion of the convergence of the method.
Finally, we conclude the analysis of the multipartite en-tanglement by considering the kicked case in the regime when the dynamics is chaotic. This system heats up to a state where all local observables on any Floquet state cor-respond to the infinite temperature values54. All
quan-tities characterizing entanglement quanquan-tities saturate to an asymptotic value at the Ehrenfest time tc
Ehr, for ev-ery initial state and field h, see Fig.S4 of the supple-mentary. The value of the QFI, being a sum of local observables, is compatible with the values of the infi-nite temperature state: fQ = 1 + N3 +O(1/N). On the other side, the entanglement entropy saturates to the value expected for a random state, which was derived by Page in [72] SP age= log m−2nm +O(1/mn) , with m, n the dimensions of the Hilbert space of the two subsys-tems and m ≤ n. In this case, for a partition of size L the dimensions are m = L + 1, n = N − L + 1 and SP age= log(L + 1) +O(1/N). This reflects on the TMI
4 10 2 100 102 104 106 108 0.1 1 10 10 5 10 4 10 3 10 2 10 1 100 0.1 1 c( t) N 2 tJ/pN quantum t4 exact DTWA classical t2
FIG. 3. The two-times regime of the square commutator in the LMG model after a quantum quench with hf = 2. In the
main plot the quantum regime of c(t) for di↵erent N : this regime starts atpN and then c(t) grows polynomially as t4.
ED results in blue for N = 20, 100, 200, 300, 400 (increasing color darkness), dashed yellow line for the polynomial fit. In the insert we show the semiclassical regime, comparing the exact c(t) with DTWA, which predicts the ⇠ t2 power-law
growth, dashed in the plot. Here N = 20 and DTWA obtained with 5 · 103 samplings.
quench dynamics, for hf hc the square commuta-tor is by terised a first semiclassical quadratic growth c(t) / t2/N3 until tr
Ehr. In this regime, semiclassical approximations describes very well the evolution of c(t) and we chose to employ DTWA. To this end we have to generalise the corresponding expression for the square commutator to the discrete phase space representation
c(t)⇠ ~ 2 N4 X i,j,k,m ⇥ xzy ij (t) yzx ij (t) ⇤ [ xzykm(t) kmyzx(t)] (4) where ↵ij (t) = ↵ i(0) @ j(t) @ i(0), with x,y,z
i the Weyl trans-form of the spin operators and the average (·) is com-puted over the initial discrete Wigner distribution [48]. At trEhr the quantum regime starts, characterised by a polynomial growth⇠ (t/N)4up to a maximum c(tr
⇤)⇠ 2. At this time, the square commutator is independent of the system size. Even DTWA, that perfectly gets mul-tipartite entanglement dynamics up to tr
rec (see Fig.1) is not able to reproduce the long time dynamics of the square commutator, see Fig.3. Indeed DTWA, despite keeping N discrete trajectories, represents all operators as factorised on each site at any time [58]. At times longer than tEhr, the operator expansion starts develop-ing longer and longer strdevelop-ings and the square commutator re-sums all the correlations, until tr
⇤, which corresponds to the time at which the string of length N occurs [48]. Quenches to hf ⌧ hcare characterized by the same time-scales tr
Ehr and tr⇤ and the same semi-classical regime 10 8 10 6 10 4 10 2 (a.) 0 0.3 0.6 0.9 (b.) 10 8 10 6 10 4 10 2 100 102 0 1 2 3 4 5 (c.) 0 0.2 0.4 0.6 0.8 1 1.2 (d.) TWA ED N = 400N = 800 t J TWA ED t J/N N = 800
FIG. 4. Square commutator dynamics (a., b.) for the LMG model past a quantum quench to the DPT point and (c., d.) for the kicked top with K = 20, hf = 2, ⌧ = 0.6. At early
times (a., c.), they are both characterized by an exponential growth up to tc
Ehr, see the right side with the log-scale on the
y axis. This regime can be perfectly reproduced by the TWA. (b., d.) At long times the behaviour is di↵erent depending on the quantum integrability properties. In (c.) the time is rescaled by N in order to show that c(t) grows up to a maximum at tr⇤. In (d.) In the Kicked top the c(t) stays constant also at very long times. In (a., c.) we used N = 800 and TWA obtained with 104 samplings
⇠ t2/N3up to tr
Ehr. Anyhow, the result at long-times is qualitatively di↵erent: the quantum regime is⇠ t3/N4 and c(tr
⇤)⇠ 10 3/N goes to zero at all times in the ther-modynamic limit. This is a direct consequence of the existence of the dynamical transition, which is detected by the scrambling [38]. Due to the presence of a macro-scopic magnetization, the support of the operators has a constrained dynamics and it will not acquire a string of length N . A special case is represented by the quench at hf = hc; despite the quantum system is integrable, we find that the square commutator grows exponentially in time up to tc
Ehras: c(t) = e2t/N3. This is due to the exis-tence of the unstable trajectory in the classical dynamics. The exponent is twice the eigenvalue of the instability matrix of the separatrix trajectory hc = 2
p
hc(1 hc) for hc= 1/2. This is valid in general for all the classical trajectories associated with DPT. To our knowledge it is the only example of an early time exponential growth in a many-body regular system. Anyhow, after tc
Ehr, c(t) keeps growing linearly in time up to the tr
⇤ and then it goes back, see Fig.4. Long-range interactions do not change drastically this analysis. In the range ↵ < 1, the early time dynamics is the same of what described be-fore. The square commutator grows like a power law at small times, even for ↵ > 2.
We conclude by considering the kicking, which induces a chaotic dynamics. As expected c(t) is initially dom-inated by the classical exponential growth, then as FIG. 8. Square commutator dynamics c(t) for the LMG model after a quantum quench to the DPT point (a., b.) and for the kicked top with K = 20, hf = 2, τ = 0.6 (c., d.). At early
times (a., c.), they are both characterized by an exponential growth up to tc
Ehr, see the right side with the log-scale on the
y axis. This regime can be perfectly reproduced by the TWA. (b., d.) At long times the behaviour is different depending on the quantum integrability properties. In (c.) the time is rescaled by N in order to show that c(t) grows up to a maximum at tr
∗. In (d.) In the Kicked top the c(t) stays
constant also at very long times. In (a., c.) we used N = 800 and TWA obtained with 104 samplings
and we find I3= log(˜n) with (19) ˜ n =(nA+ 1)(nB+ 1)(nC+ 1)(nA+ nB+ nC+ 1) (nA+ nB+ 1)(nA+ nC+ 1)(nB+ nC+ 1) . B. Scrambling
Scrambling, as measured by the square commutator, be-haves in a profoundly different way from entanglement. It is characterised by an initial semi-classical regime and a second quantum non-perturbative growth. Interestingly, this phenomenon is very evident in the regular regime (Fig.7), and it is much less clear in the chaotic one (Fig.8) already discussed at the beginning of the paper.
In the case of the quench dynamics, for hf hc the square commutator is characterised by a first semiclas-sical quadratic growth c(t)∝ t2/N3 until tr
Ehr. In this regime, semiclassical approximations describe very well the evolution of c(t) and we chose to employ DTWA. To this end, we generalised the corresponding expression for the square commutator to the discrete phase space representation c(t)∼ ~ 2 N4 X i,j,k,m δijxzy(t)− δ yzx ij (t) [δxzykm(t)− δkmyzx(t)] (20)
